Beam-Columns

Design of members subjected to combined axial compression and flexural loads.
Most structural members are subjected to both axial forces and bending moments simultaneously. A column in a rigid frame resists axial gravity loads but also must resist bending moments transferred from the beams or caused by lateral wind/seismic forces. When the axial load is significant, the member is classified as a beam-column. The design of beam-columns involves checking interaction equations that account for stability and strength under combined loading.

Combined Forces

The interaction between axial compression and flexure reduces the capacity of the member compared to either force acting alone. The interaction is generally non-linear, especially for slender members, because the bending moments exacerbate the column's tendency to buckle, and the axial load increases the bending stresses (second-order effects).

Interaction Equations (AISC Chapter H)

AISC provides two interaction equations depending on the ratio of the required axial strength (PuP_u) to the available axial strength (PcP_c).

Case 1: High Axial Load ($P_u / P_c \ge 0.2$)

If the axial load is dominant (typical for lower-story columns):
PuPc+89(MuxMcx+MuyMcy)≤1.0\frac{P_u}{P_c} + \frac{8}{9} \left( \frac{M_{ux}}{M_{cx}} + \frac{M_{uy}}{M_{cy}} \right) \le 1.0

Case 2: Low Axial Load ($P_u / P_c < 0.2$)

If the bending moment is dominant (typical for roof beams or upper-story columns):
Pu2Pc+(MuxMcx+MuyMcy)≤1.0\frac{P_u}{2 P_c} + \left( \frac{M_{ux}}{M_{cx}} + \frac{M_{uy}}{M_{cy}} \right) \le 1.0
Where:
  • PuP_u = Required axial strength (factored load)
  • PcP_c = Available axial strength (ϕcPn\phi_c P_n) determined according to Chapter E (Compression)
  • Mux,MuyM_{ux}, M_{uy} = Required flexural strengths about x and y axes (factored moments, including second-order effects)
  • Mcx,McyM_{cx}, M_{cy} = Available flexural strengths (ϕbMn\phi_b M_n) about x and y axes determined according to Chapter F (Flexure)

Beam-Column Interaction (AISC Chapter H)

Member Capacities (Assumed W14x90):

  • Axial Compressive Strength (φP_c): 1000 kips
  • Flexural Strength (φM_cx): 600 k-ft
200 kips
150 k-ft

Interaction Value: 0.422

Equation H1-1a used (Pu/φPc ≥ 0.2)

Loading chart...
Limit: ≤ 1.0

Second-Order Effects and Stability Analysis

Accounting for the additional moments caused by deformation.
In beam-columns, the axial load acts on the deflected shape caused by the moment, creating additional internal moments. These are called second-order effects. First-order analysis (traditional structural analysis) assumes the geometry of the structure remains unchanged under load. AISC requires that stability analysis must account for these second-order effects.

P-Delta (Δ\Delta) and P-delta (δ\delta)

  • P−δP-\delta (Small Delta): The effect of member curvature between brace points. The axial load PP acts on the local deflection δ\delta of the member itself, increasing the internal moment along the span.
  • P−ΔP-\Delta (Large Delta): The effect of lateral drift of the entire frame (sway). The total gravity load PP on the story acts on the lateral displacement Δ\Delta of the story relative to its base, creating a massive overturning moment that must be resisted by the columns and connections.

Analysis Methods

Three primary methods for capturing second-order effects in AISC.

1. Direct Analysis Method (DAM) - AISC Chapter C

The modern, preferred approach in AISC. It explicitly accounts for all stability effects in the structural analysis model itself, rather than relying on empirical KK-factors.
It requires:
  1. Second-Order Analysis: The computer model must rigorously calculate both P−ΔP-\Delta and P−δP-\delta effects.
  2. Notional Loads: Small lateral loads (0.002Yi0.002 Y_i) are applied to account for initial geometric imperfections (out-of-plumbness).
  3. Reduced Stiffness: The flexural and axial stiffness of members contributing to stability are reduced (0.80EI0.80 EI and 0.80EA0.80 EA) to account for the softening effect of residual stresses and partial yielding.
Because stability is handled directly in the analysis, the effective length factor (KK) is simply taken as 1.01.0 for all members when calculating the available axial strength (PcP_c) for the interaction equations.

2. Effective Length Method (ELM) - AISC Appendix 7

The traditional approach. It relies on determining the effective length factor (KK) using alignment charts (nomographs), which can be complex for sway frames (K>1.0K > 1.0).
It is only permitted when the ratio of second-order drift to first-order drift (Δ2nd/Δ1st\Delta_{2nd} / \Delta_{1st}) is not overly large (less than 1.5). It does not require stiffness reductions, but it requires accurate calculation of KK factors based on relative joint stiffness (GG-factors).

3. First-Order Analysis Method - AISC Appendix 7

A simplified version of DAM, only applicable when the structure is relatively stiff against lateral drift (Δ2nd/Δ1st≤1.5\Delta_{2nd} / \Delta_{1st} \le 1.5). It uses a first-order analysis but applies much larger notional loads (0.004Yi0.004 Y_i or more) and assumes K=1.0K=1.0.

The Direct Analysis Method (DAM)

The modern AISC standard for stability design, superseding older effective length methods.
Historically, engineers accounted for frame stability using the Effective Length Method (ELM), which relies on alignment charts to calculate KK-factors. However, ELM has significant limitations when dealing with highly inelastic frames or frames with significant asymmetry.
AISC Chapter C now mandates the Direct Analysis Method (DAM) as the primary approach for stability design. The DAM explicitly models the imperfections and inelasticity directly in the structural analysis, rather than trying to account for them implicitly with KK-factors.

Key Requirements of the DAM

When using the Direct Analysis Method, the effective length factor (KK) for all columns is simply taken as K=1.0K = 1.0. Instead of adjusting KK, the method requires three major adjustments to the structural analysis itself:
  • Rigorous Second-Order Analysis: The analysis must explicitly capture both P−ΔP-\Delta (sway) and P−δP-\delta (member curvature) effects.
  • Notional Loads (NiN_i): To account for initial out-of-plumbness of the building framework, small lateral loads (Ni=0.002YiN_i = 0.002 Y_i, where YiY_i is the gravity load) are applied at each framing level.
  • Reduced Stiffness: To account for the loss of stiffness due to residual stresses and partial yielding, the flexural stiffness (EIEI) and axial stiffness (EAEA) of all members contributing to stability are reduced. Typically, 0.8EA0.8EA and 0.8τbEI0.8\tau_b EI are used in the computer model.
By applying notional loads and reducing the frame's stiffness, the structural analysis directly generates the amplified moments and forces. The member capacities are then simply checked using K=1.0K=1.0, drastically simplifying the design of complex beam-columns in sway frames.

Moment Amplification Method (AISC Appendix 8)

An approximate method to calculate second-order moments from first-order results.
If a rigorous second-order computer analysis is not available, the required moment MuM_u (and axial load PuP_u) can be determined by amplifying the results of a simple first-order elastic analysis.

AISC Moment Amplification Method

Calculates the amplified bending moment accounting for second-order P-Delta and P-delta effects.

$$ M_r = B_1 M_{nt} + B_2 M_{lt} $$

Amplification Factor (B1B_1)

Accounts for the moment gradient and the Euler buckling load of the individual column.
B1=Cm1−αPrPe1≥1.0B_1 = \frac{C_m}{1 - \frac{\alpha P_r}{P_{e1}}} \ge 1.0
Where:
  • α=1.0\alpha = 1.0 (LRFD) or 1.61.6 (ASD).
  • PrP_r = Required axial compressive strength.
  • CmC_m = Coefficient based on moment gradient (typically 0.6−0.4(M1/M2)0.6 - 0.4(M_1/M_2) for members without transverse loads between supports). If the member is bent in reverse curvature, CmC_m is small. If bent in single curvature, CmC_m approaches 1.0.
  • Pe1P_{e1} = Euler buckling load in the plane of bending (π2EI/(K1L)2\pi^2 EI / (K_1 L)^2).

Amplification Factor (B2B_2)

Accounts for the story drift and the total gravity load on the story.
B2=11−αΣPntΣPe2≥1.0B_2 = \frac{1}{1 - \frac{\alpha \Sigma P_{nt}}{\Sigma P_{e2}}} \ge 1.0
Where:
  • ΣPnt\Sigma P_{nt} = Total required vertical load on all columns in the story.
  • ΣPe2\Sigma P_{e2} = Total elastic sway buckling strength of the story.

Bracing Requirements (AISC Appendix 6)

Providing adequate stability to columns, beams, and frames.
For the equations and design capacities to be valid, structural members must be adequately braced against buckling. A brace must provide both sufficient strength (to resist the force of the member trying to buckle) and stiffness (to prevent the member from moving).

Types of Bracing

  • Relative Bracing: Controls the relative movement between two adjacent brace points (e.g., diagonal cross-bracing in a frame, or a shear wall). It is generally stiffer and stronger because it acts globally on the system.
  • Nodal Bracing: Restrains lateral movement independently at a specific point on a member (e.g., a floor beam framing perpendicularly into the weak axis of a column). The brace must be attached to a rigid support.
AISC Appendix 6 provides specific required stiffness (βbr\beta_{br}) and required strength (PbrP_{br}) formulas for column bracing, beam lateral bracing, and beam torsional bracing to ensure they function correctly and prevent premature buckling. If a brace is not stiff enough, the column will buckle right through the brace point.
Key Takeaways
  • Beam-columns must be designed to withstand both axial compression and bending moments simultaneously, checked via AISC interaction equations.
  • AISC provides two interaction equations: one for when axial load governs (Pu/Pc≥0.2P_u/P_c \ge 0.2) and one for when bending governs.
  • Second-order effects (P−δP-\delta member curvature and P−ΔP-\Delta frame sway) must be included in the moment calculations because axial loads increase bending moments on deflected shapes.
  • The Direct Analysis Method (DAM) is the modern, preferred approach, applying K=1.0K=1.0 by directly modeling initial imperfections (notional loads) and reduced stiffness in a rigorous second-order computer analysis.
  • The Moment Amplification Method uses factors (B1B_1 and B2B_2) to artificially increase the first-order design moments to account for second-order effects manually.
  • Bracing systems must be designed for both adequate strength and stiffness to effectively prevent buckling.
PreviousBeams: Shear and Serviceability - Examples & Applications
Quiz Me
NextBeam-Columns - Examples & Applications